Finite Group Databasebeta

← #27144351

Subgroups — #27144351

Order 12416.

600 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690768C211yesmaximal, characteristic
2C2×#15690767C211yesmaximal, characteristic
3C2×C4⋊C4×D97C211yesmaximal, characteristic
4#20260634C211yesmaximal, characteristic, complemented
5#20260633C211yesmaximal, characteristic, complemented
6#20260632C212yesmaximal, complemented
7#20260631C211yesmaximal, characteristic, complemented
8#20260630C212yesmaximal, complemented
9#20260594C212yesmaximal
10#20260592C212yesmaximal
11#389×C97C211yesmaximal, characteristic, complemented
12C22×C4×D97C2212yes—
13C22×C4×D97C2211yescharacteristic
14#15690768—24no—
15#15690767—24no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×C4⋊Dic97C2212yes—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2212yes—
22C2×#13095490C2212yes—
23C2×#13095490C2212yes—
24C4⋊C4×D97C418yescomplemented
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2212yes—
28C2×C4×Dic97C2212yes—
29C2×C4×Dic97C2211yescharacteristic
30#15690717C2212yes—
31#15690717C2212yes—
32#15690717C2212yes—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2212yes—
35C2×C4⋊C4×C97C2211yescharacteristic
36C2×C42×C97C2211yescharacteristic
37#95×C97C2212yes—
38C23×D97C2311yescharacteristic
39C22×Dic97C2311yescharacteristic
40C22×Dic97C2311yescharacteristic
41C22×Dic97C2312yes—
42C22×Dic97C2312yes—
43C22×Dic97C2311yescharacteristic
44C2×C4×D97—42no—
45C2×C4×D97—42no—
46C2×C4×D97C2×C418yes—
47C2×C4×D97C2×C414yes—
48C2×C4×D97D414yes—
49C2×C4×D97Q814yes—
50C4⋊Dic97—22no—
51C4⋊Dic97—42no—
52C4⋊Dic97C2×C414yes—
53#13095490—22no—
54#13095490—22no—
55#13095490—42no—
56#13095490—42no—
57#13095490C2×C418yes—
58#13095488—22no—
59#13095488—22no—
60#13095488—24no—
61C4×Dic97—22no—
62C4×Dic97—24no—
63C22×C4×C97C2312yes—
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C22×C4×C97C2312yes—
67C22×C4×C97C2311yescharacteristic
68C4⋊C4×C97—22no—
69C4⋊C4×C97—42no—
70C4⋊C4×C97C2×C414yes—
71C42×C97—22no—
72C23×C97C2411yescharacteristic
73C22×D97—21no—
74C22×D97—21no—
75C22×D97—21no—
76C22×D97—21no—
77C22×D97C22×C412yes—
78C22×D97C2×D412yes—
79C22×D97C2×Q812yes—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—21no—
83C2×C4×C97—21no—
84C2×C4×C97—22no—
85C2×C4×C97—21no—
86C2×C4×C97—22no—
87C2×C4×C97—21no—
88C2×C4×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97C22×C414yes—
92C2×C4×C97C22×C412yes—
93C2×C4×C97C2×D412yes—
94C2×C4×C97C2×Q812yes—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—22no—
100C2×Dic97—21no—
101C2×Dic97—22no—
102C2×Dic97—21no—
103C2×Dic97—21no—
104C2×Dic97—21no—
105C2×Dic97—22no—
106C2×Dic97C22×C414yes—
107C2×Dic97C22×C412yes—
108C2×Dic97C2×D412yes—
109C2×Dic97C2×Q812yes—
110C4×D97—48no—
111C4×D97C4⋊C4116yescomplemented
112C22×C97C23×C411yescharacteristic
113C22×C97C22×D411yescharacteristic
114C22×C97C22×Q811yescharacteristic
115C22×C97H(5,𝔽2)11yescharacteristic
116C22×C97H(5,𝔽2)11yescharacteristic
117C22×C972−1+411yescharacteristic
118C22×C972−1+411yescharacteristic
119C2×D97—22no—
120C2×D97—22no—
121C2×D97—22no—
122C2×D97—22no—
123C2×D97—22no—
124C2×D97—22no—
125C2×D97C2×C4⋊C414yes—
126C4×C97—24no—
127C4×C97—41no—
128C4×C97—42no—
129C4×C97—41no—
130C4×C97C2×C4⋊C414yes—
131Dic97—24no—
132Dic97—41no—
133Dic97—41no—
134Dic97—42no—
135Dic97C2×C4⋊C414yes—
136C2×C97C22×C4⋊C411yescharacteristic
137C2×C97#51811yescharacteristic
138C2×C97#51911yescharacteristic
139C2×C97#53711yescharacteristic
140C2×C97#53711yescharacteristic
141C2×C97#54411yescharacteristic
142C2×C97#54411yescharacteristic
143D97—24no—
144#2316—971nomaximal, Sylow
145C97#231611yescharacteristic, complemented, Sylow
146C2×#117—971no—
147C2×#117—971no—
148C22×C4⋊C4—971no—
149#389—972no—
150#389—971no—
151#389—971no—
152#389—971no—
153#389—972no—
154#389D9711yescharacteristic, complemented
155#380—972no—
156#380—972no—
157C23×C4—971no—
158C23×C4—972no—
159#117—1944no—
160#117—1944no—
161C2×C4⋊C4—972no—
162C2×C4⋊C4—971no—
163C2×C4⋊C4—978no—
164C2×C4⋊C4—972no—
165C2×C4⋊C4—972no—
166C2×C4⋊C4—972no—
167C2×C4⋊C4—971no—
168C2×C4⋊C4—971no—
169C2×C4⋊C4—971no—
170C2×C4⋊C4C2×D9711yescharacteristic
171C2×C4⋊C4C2×D9712yes—
172C2×C4⋊C4C2×D9711yescharacteristic
173C2×C22⋊C4—971no—
174C2×C22⋊C4—972no—
175C2×C22⋊C4—971no—
176C2×C42—972no—
177C2×C42—971no—
178C2×C42C2×D9711yescharacteristic
179#95—972no—
180#95—972no—
181#95—972no—
182#95C2×D9712yes—
183C24—971no—
184C22×C4—971no—
185C22×C4—974no—
186C22×C4—974no—
187C22×C4—974no—
188C22×C4—972no—
189C22×C4—978no—
190C22×C4—971no—
191C22×C4—972no—
192C22×C4—971no—
193C22×C4—3882no—
194C22×C4—3882no—
195C22×C4C22×D9711yescharacteristic
196C22×C4C22×D9712yes—
197C22×C4C22×D9711yescharacteristic
198C22×C4C22×D9712yes—
199C22×C4C22×D9711yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—978no—
203C4⋊C4—974no—
204C4⋊C4—3882no—
205C4⋊C4—3882no—
206C4⋊C4—3882no—
207C4⋊C4—1942no—
208C4⋊C4—1942no—
209C4⋊C4—1942no—
210C4⋊C4C4×D9714yes—
211C22⋊C4—1942no—
212C22⋊C4—1942no—
213C22⋊C4—1944no—
214C42—22no—
215C42—1944no—
216C42—1942no—
217C23—972no—
218C23—972no—
219C23—972no—
220C23—1941no—
221C23—1941no—
222C23—1941no—
223C23—1941no—
224C23C23×D9711yescharacteristic
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—972no—
237C2×C4—972no—
238C2×C4—972no—
239C2×C4—9716no—
240C2×C4—974no—
241C2×C4—3888no—
242C2×C4—1941no—
243C2×C4—1942no—
244C2×C4—1941no—
245C2×C4—1941no—
246C2×C4—1942no—
247C2×C4—1941no—
248C2×C4—1941no—
249C2×C4—1942no—
250C2×C4—1941no—
251C2×C4—1942no—
252C2×C4—1942no—
253C2×C4C2×C4×D9714yes—
254C2×C4C2×C4×D9712yes—
255C2×C4D4×D9712yes—
256C2×C4Q8×D9712yes—
257C22—974no—
258C22—1942no—
259C22—1942no—
260C22—1942no—
261C22—1942no—
262C22—1942no—
263C22—1942no—
264C22C22×C4×D9711yescharacteristic
265C22C2×D4×D9711yescharacteristic
266C22C2×Q8×D9711yescharacteristic
267C22#1569085511yescharacteristic
268C22#1569085611yescharacteristic
269C22#1569085711yescharacteristic
270C22#1569085811yescharacteristic
271C4—24no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—974no—
276C4—3881no—
277C4—3882no—
278C4—3881no—
279C4—1944no—
280C4C4⋊C4×D9714yes—
281C2—1944no—
282C2C2×C4⋊C4×D9711yescharacteristic
283C2#2026120711yescharacteristic
284C2#2026121111yescharacteristic
285C2#2026131411yescharacteristic
286C2#2026131511yescharacteristic
287C2#2026136211yescharacteristic
288C2#2026136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#389×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#389D97
hypocenterC97#2316
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #389

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20260634C2semidirect
5. #20260633C2semidirect
6. #20260632C2semidirect
7. #20260631C2semidirect
8. #20260630C2semidirect
11. #389×C97C2semidirect
24. C4⋊C4×D97C4semidirect
111. C4×D97C4⋊C4semidirect
145. C97#2316semidirect
154. #389D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156907682
C2×#156907672
C2×C4⋊C4×D972
#202606342
#202606332
#202606322
#202606312
#202606302
#202605942
#202605922
#389×C972
#231697